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Series Partial Sum Sequence
In order to give a meaning to the sum of an infinite series Σun, we form a sequence of partial sums.
Let
be any given series, where the terms may be positive or negative.
Let Sn = u1 + u2 + …. + un, be the sum of first n terms of the series Σun. Then Sn is called the partial sum of the first n terms of the given series and the sequence <Sn>, where Sn = u1 + u2 + …. + un, ∀ n ϵ N, is called the sequence of the partial sums of given series
.
If Sn is known, we can find un and hence the series,
for un = Sn – Sn-1
∴ From the above discussion, we conclude that a sequence <Sn> of partial sums is associated to each series
and vice versa.
∴ We can assign the same behaviour to the series
as is exhibited by its sequence <Sn> of partial sums as
.
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Let
Let Sn = u1 + u2 + …. + un, be the sum of first n terms of the series Σun. Then Sn is called the partial sum of the first n terms of the given series and the sequence <Sn>, where Sn = u1 + u2 + …. + un, ∀ n ϵ N, is called the sequence of the partial sums of given series
If Sn is known, we can find un and hence the series,
for un = Sn – Sn-1
∴ From the above discussion, we conclude that a sequence <Sn> of partial sums is associated to each series
∴ We can assign the same behaviour to the series
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